3.226 \(\int f^{\frac{c}{(a+b x)^2}} x^2 \, dx\)

Optimal. Leaf size=206 \[ -\frac{\sqrt{\pi } a^2 \sqrt{c} \sqrt{\log (f)} \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{b^3}+\frac{a^2 (a+b x) f^{\frac{c}{(a+b x)^2}}}{b^3}-\frac{2 \sqrt{\pi } c^{3/2} \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{3 b^3}+\frac{a c \log (f) \text{Ei}\left (\frac{c \log (f)}{(a+b x)^2}\right )}{b^3}+\frac{(a+b x)^3 f^{\frac{c}{(a+b x)^2}}}{3 b^3}-\frac{a (a+b x)^2 f^{\frac{c}{(a+b x)^2}}}{b^3}+\frac{2 c \log (f) (a+b x) f^{\frac{c}{(a+b x)^2}}}{3 b^3} \]

[Out]

(a^2*f^(c/(a + b*x)^2)*(a + b*x))/b^3 - (a*f^(c/(a + b*x)^2)*(a + b*x)^2)/b^3 + (f^(c/(a + b*x)^2)*(a + b*x)^3
)/(3*b^3) - (a^2*Sqrt[c]*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Sqrt[Log[f]])/b^3 + (2*c*f^(c/(a + b*
x)^2)*(a + b*x)*Log[f])/(3*b^3) + (a*c*ExpIntegralEi[(c*Log[f])/(a + b*x)^2]*Log[f])/b^3 - (2*c^(3/2)*Sqrt[Pi]
*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Log[f]^(3/2))/(3*b^3)

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Rubi [A]  time = 0.214925, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2226, 2206, 2211, 2204, 2214, 2210} \[ -\frac{\sqrt{\pi } a^2 \sqrt{c} \sqrt{\log (f)} \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{b^3}+\frac{a^2 (a+b x) f^{\frac{c}{(a+b x)^2}}}{b^3}-\frac{2 \sqrt{\pi } c^{3/2} \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{3 b^3}+\frac{a c \log (f) \text{Ei}\left (\frac{c \log (f)}{(a+b x)^2}\right )}{b^3}+\frac{(a+b x)^3 f^{\frac{c}{(a+b x)^2}}}{3 b^3}-\frac{a (a+b x)^2 f^{\frac{c}{(a+b x)^2}}}{b^3}+\frac{2 c \log (f) (a+b x) f^{\frac{c}{(a+b x)^2}}}{3 b^3} \]

Antiderivative was successfully verified.

[In]

Int[f^(c/(a + b*x)^2)*x^2,x]

[Out]

(a^2*f^(c/(a + b*x)^2)*(a + b*x))/b^3 - (a*f^(c/(a + b*x)^2)*(a + b*x)^2)/b^3 + (f^(c/(a + b*x)^2)*(a + b*x)^3
)/(3*b^3) - (a^2*Sqrt[c]*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Sqrt[Log[f]])/b^3 + (2*c*f^(c/(a + b*
x)^2)*(a + b*x)*Log[f])/(3*b^3) + (a*c*ExpIntegralEi[(c*Log[f])/(a + b*x)^2]*Log[f])/b^3 - (2*c^(3/2)*Sqrt[Pi]
*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Log[f]^(3/2))/(3*b^3)

Rule 2226

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rule 2206

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[((c + d*x)*F^(a + b*(c + d*x)^n))/d, x]
- Dist[b*n*Log[F], Int[(c + d*x)^n*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n]
&& ILtQ[n, 0]

Rule 2211

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[1/(d*(m + 1))
, Subst[Int[F^(a + b*x^2), x], x, (c + d*x)^(m + 1)], x] /; FreeQ[{F, a, b, c, d, m, n}, x] && EqQ[n, 2*(m + 1
)]

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2214

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), Int[(c + d*x)^(m + n)*F^(a + b*(c +
d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rule 2210

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[(F^a*ExpIntegralEi[
b*(c + d*x)^n*Log[F]])/(f*n), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int f^{\frac{c}{(a+b x)^2}} x^2 \, dx &=\int \left (\frac{a^2 f^{\frac{c}{(a+b x)^2}}}{b^2}-\frac{2 a f^{\frac{c}{(a+b x)^2}} (a+b x)}{b^2}+\frac{f^{\frac{c}{(a+b x)^2}} (a+b x)^2}{b^2}\right ) \, dx\\ &=\frac{\int f^{\frac{c}{(a+b x)^2}} (a+b x)^2 \, dx}{b^2}-\frac{(2 a) \int f^{\frac{c}{(a+b x)^2}} (a+b x) \, dx}{b^2}+\frac{a^2 \int f^{\frac{c}{(a+b x)^2}} \, dx}{b^2}\\ &=\frac{a^2 f^{\frac{c}{(a+b x)^2}} (a+b x)}{b^3}-\frac{a f^{\frac{c}{(a+b x)^2}} (a+b x)^2}{b^3}+\frac{f^{\frac{c}{(a+b x)^2}} (a+b x)^3}{3 b^3}+\frac{(2 c \log (f)) \int f^{\frac{c}{(a+b x)^2}} \, dx}{3 b^2}-\frac{(2 a c \log (f)) \int \frac{f^{\frac{c}{(a+b x)^2}}}{a+b x} \, dx}{b^2}+\frac{\left (2 a^2 c \log (f)\right ) \int \frac{f^{\frac{c}{(a+b x)^2}}}{(a+b x)^2} \, dx}{b^2}\\ &=\frac{a^2 f^{\frac{c}{(a+b x)^2}} (a+b x)}{b^3}-\frac{a f^{\frac{c}{(a+b x)^2}} (a+b x)^2}{b^3}+\frac{f^{\frac{c}{(a+b x)^2}} (a+b x)^3}{3 b^3}+\frac{2 c f^{\frac{c}{(a+b x)^2}} (a+b x) \log (f)}{3 b^3}+\frac{a c \text{Ei}\left (\frac{c \log (f)}{(a+b x)^2}\right ) \log (f)}{b^3}-\frac{\left (2 a^2 c \log (f)\right ) \operatorname{Subst}\left (\int f^{c x^2} \, dx,x,\frac{1}{a+b x}\right )}{b^3}+\frac{\left (4 c^2 \log ^2(f)\right ) \int \frac{f^{\frac{c}{(a+b x)^2}}}{(a+b x)^2} \, dx}{3 b^2}\\ &=\frac{a^2 f^{\frac{c}{(a+b x)^2}} (a+b x)}{b^3}-\frac{a f^{\frac{c}{(a+b x)^2}} (a+b x)^2}{b^3}+\frac{f^{\frac{c}{(a+b x)^2}} (a+b x)^3}{3 b^3}-\frac{a^2 \sqrt{c} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right ) \sqrt{\log (f)}}{b^3}+\frac{2 c f^{\frac{c}{(a+b x)^2}} (a+b x) \log (f)}{3 b^3}+\frac{a c \text{Ei}\left (\frac{c \log (f)}{(a+b x)^2}\right ) \log (f)}{b^3}-\frac{\left (4 c^2 \log ^2(f)\right ) \operatorname{Subst}\left (\int f^{c x^2} \, dx,x,\frac{1}{a+b x}\right )}{3 b^3}\\ &=\frac{a^2 f^{\frac{c}{(a+b x)^2}} (a+b x)}{b^3}-\frac{a f^{\frac{c}{(a+b x)^2}} (a+b x)^2}{b^3}+\frac{f^{\frac{c}{(a+b x)^2}} (a+b x)^3}{3 b^3}-\frac{a^2 \sqrt{c} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right ) \sqrt{\log (f)}}{b^3}+\frac{2 c f^{\frac{c}{(a+b x)^2}} (a+b x) \log (f)}{3 b^3}+\frac{a c \text{Ei}\left (\frac{c \log (f)}{(a+b x)^2}\right ) \log (f)}{b^3}-\frac{2 c^{3/2} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right ) \log ^{\frac{3}{2}}(f)}{3 b^3}\\ \end{align*}

Mathematica [A]  time = 0.103422, size = 131, normalized size = 0.64 \[ \frac{-\sqrt{\pi } \sqrt{c} \sqrt{\log (f)} \left (3 a^2+2 c \log (f)\right ) \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )+b x f^{\frac{c}{(a+b x)^2}} \left (b^2 x^2+2 c \log (f)\right )+3 a c \log (f) \text{Ei}\left (\frac{c \log (f)}{(a+b x)^2}\right )}{3 b^3}+\frac{a \left (a^2+2 c \log (f)\right ) f^{\frac{c}{(a+b x)^2}}}{3 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(c/(a + b*x)^2)*x^2,x]

[Out]

(a*f^(c/(a + b*x)^2)*(a^2 + 2*c*Log[f]))/(3*b^3) + (3*a*c*ExpIntegralEi[(c*Log[f])/(a + b*x)^2]*Log[f] - Sqrt[
c]*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Sqrt[Log[f]]*(3*a^2 + 2*c*Log[f]) + b*f^(c/(a + b*x)^2)*x*(
b^2*x^2 + 2*c*Log[f]))/(3*b^3)

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Maple [A]  time = 0.04, size = 175, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{3}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}+{\frac{{a}^{3}}{3\,{b}^{3}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}+{\frac{2\,c\ln \left ( f \right ) x}{3\,{b}^{2}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}+{\frac{2\,ac\ln \left ( f \right ) }{3\,{b}^{3}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}-{\frac{2\, \left ( \ln \left ( f \right ) \right ) ^{2}{c}^{2}\sqrt{\pi }}{3\,{b}^{3}}{\it Erf} \left ({\frac{1}{bx+a}\sqrt{-c\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}-{\frac{{a}^{2}c\ln \left ( f \right ) \sqrt{\pi }}{{b}^{3}}{\it Erf} \left ({\frac{1}{bx+a}\sqrt{-c\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}-{\frac{ac\ln \left ( f \right ) }{{b}^{3}}{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{ \left ( bx+a \right ) ^{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c/(b*x+a)^2)*x^2,x)

[Out]

1/3*f^(c/(b*x+a)^2)*x^3+1/3/b^3*a^3*f^(c/(b*x+a)^2)+2/3/b^2*ln(f)*c*f^(c/(b*x+a)^2)*x+2/3/b^3*ln(f)*c*f^(c/(b*
x+a)^2)*a-2/3/b^3*ln(f)^2*c^2*Pi^(1/2)/(-c*ln(f))^(1/2)*erf((-c*ln(f))^(1/2)/(b*x+a))-1/b^3*a^2*ln(f)*c*Pi^(1/
2)/(-c*ln(f))^(1/2)*erf((-c*ln(f))^(1/2)/(b*x+a))-1/b^3*a*ln(f)*c*Ei(1,-c*ln(f)/(b*x+a)^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (b^{2} x^{3} + 2 \, c x \log \left (f\right )\right )} f^{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{3 \, b^{2}} - \int \frac{2 \,{\left (3 \, a b^{2} c x^{2} \log \left (f\right ) + a^{3} c \log \left (f\right ) +{\left (3 \, a^{2} b c \log \left (f\right ) - 2 \, b c^{2} \log \left (f\right )^{2}\right )} x\right )} f^{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{3 \,{\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c/(b*x+a)^2)*x^2,x, algorithm="maxima")

[Out]

1/3*(b^2*x^3 + 2*c*x*log(f))*f^(c/(b^2*x^2 + 2*a*b*x + a^2))/b^2 - integrate(2/3*(3*a*b^2*c*x^2*log(f) + a^3*c
*log(f) + (3*a^2*b*c*log(f) - 2*b*c^2*log(f)^2)*x)*f^(c/(b^2*x^2 + 2*a*b*x + a^2))/(b^5*x^3 + 3*a*b^4*x^2 + 3*
a^2*b^3*x + a^3*b^2), x)

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Fricas [A]  time = 1.8878, size = 312, normalized size = 1.51 \begin{align*} \frac{3 \, a c{\rm Ei}\left (\frac{c \log \left (f\right )}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) \log \left (f\right ) + \sqrt{\pi }{\left (3 \, a^{2} b + 2 \, b c \log \left (f\right )\right )} \sqrt{-\frac{c \log \left (f\right )}{b^{2}}} \operatorname{erf}\left (\frac{b \sqrt{-\frac{c \log \left (f\right )}{b^{2}}}}{b x + a}\right ) +{\left (b^{3} x^{3} + a^{3} + 2 \,{\left (b c x + a c\right )} \log \left (f\right )\right )} f^{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{3 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c/(b*x+a)^2)*x^2,x, algorithm="fricas")

[Out]

1/3*(3*a*c*Ei(c*log(f)/(b^2*x^2 + 2*a*b*x + a^2))*log(f) + sqrt(pi)*(3*a^2*b + 2*b*c*log(f))*sqrt(-c*log(f)/b^
2)*erf(b*sqrt(-c*log(f)/b^2)/(b*x + a)) + (b^3*x^3 + a^3 + 2*(b*c*x + a*c)*log(f))*f^(c/(b^2*x^2 + 2*a*b*x + a
^2)))/b^3

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(c/(b*x+a)**2)*x**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{\frac{c}{{\left (b x + a\right )}^{2}}} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c/(b*x+a)^2)*x^2,x, algorithm="giac")

[Out]

integrate(f^(c/(b*x + a)^2)*x^2, x)